| L(s) = 1 | + 2·3-s − 3·5-s + 2·7-s + 3·9-s − 11-s + 5·13-s − 6·15-s + 2·17-s + 9·19-s + 4·21-s − 9·23-s + 25-s + 4·27-s + 4·29-s − 2·31-s − 2·33-s − 6·35-s + 2·37-s + 10·39-s − 21·41-s + 11·43-s − 9·45-s + 16·47-s + 6·49-s + 4·51-s − 6·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s + 0.755·7-s + 9-s − 0.301·11-s + 1.38·13-s − 1.54·15-s + 0.485·17-s + 2.06·19-s + 0.872·21-s − 1.87·23-s + 1/5·25-s + 0.769·27-s + 0.742·29-s − 0.359·31-s − 0.348·33-s − 1.01·35-s + 0.328·37-s + 1.60·39-s − 3.27·41-s + 1.67·43-s − 1.34·45-s + 2.33·47-s + 6/7·49-s + 0.560·51-s − 0.824·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.599064007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.599064007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 21 T + 188 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 30 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.391825225555347839674702444050, −9.223728842024629669379815413301, −8.468722088567413527148484653861, −8.420581200988662940973264913117, −7.938846026687727681812166154136, −7.83947958871168757183606892632, −7.39940880743821438983841520542, −7.01470206922185462699999256620, −6.46531196120207072328983746704, −5.90661702941994201242035115756, −5.32243873079359362648755579691, −5.14276418146773270152626038242, −4.35629189196835752454266510685, −3.90797758386065578451448703153, −3.63540500566311422323770437028, −3.42286352410900650761091804677, −2.60514174374468292176840762071, −2.11553242144387370847673920882, −1.34515202813911388997662488240, −0.75528180488277671043780057674,
0.75528180488277671043780057674, 1.34515202813911388997662488240, 2.11553242144387370847673920882, 2.60514174374468292176840762071, 3.42286352410900650761091804677, 3.63540500566311422323770437028, 3.90797758386065578451448703153, 4.35629189196835752454266510685, 5.14276418146773270152626038242, 5.32243873079359362648755579691, 5.90661702941994201242035115756, 6.46531196120207072328983746704, 7.01470206922185462699999256620, 7.39940880743821438983841520542, 7.83947958871168757183606892632, 7.938846026687727681812166154136, 8.420581200988662940973264913117, 8.468722088567413527148484653861, 9.223728842024629669379815413301, 9.391825225555347839674702444050
